The math related to investment diversification

may be my maths (its Friday and i have had a beer)but for example -
£12000 as 12 equal loans and one defaults in year one the £11000 would give you a flat £1320 return(no reinvestment,amortized etc)  more years go past you would make even less of a loss due to the interest on the 12th loan and get a bigger return
While its good to know what percentage of loans default its still only an average which you could better or may be hit a bad spell.So still think a good spread is important 

Mathematically speaking diversification is inversely related to the square root of N (assuming low correlations between investments). What this basically means is to halve the risk you have to Quadruple the number of investments. So if you have 1 investment to halve the risk you have to go to 4 investments. If you have 1000 investments to halve the risk you have to go to 4000 investments. You could definitely say the more loans the better, but there is a diminishing effect - and remember you are also diversifying the upside as well as the down side. I have always taken the view you are better investing in quality rather than quantity.

no not a worthwhile risk return(but still better then a cash ISA )but was based on your 12 loans with one default which is why most people have a lot more loans to spread the risk.
Some one like Lending works spreads your money across a minimum of a hundred loans to get 4.8% on a 5 year account. Hopefully we should be able to run a self invested product on higher rate loans in the same way(assuming you hold the loans to term which many people do not) 
 

Hi Justme

You are thinking about this absolutely correctly. Diversification does not reduce the underlying default risk of any loan, or change the expected return profile. It reduces the risk that your losses from defaults will exceed a certain level (i.e if you buy a single loan you could have a 100% loss, that's impossible if you have 1,000 loans). The real benefits of diversification come from spreading your investments across loans in different countries, different lenders, and different types of loans (secured, unsecured, buyback guaranteed etc). It still doesn't reduce the default risk on any individual loan, but it protects against a huge impact if a lender/platform blows up, or a single country has major economic problems etc. 

This is not specific to loans. Take a simple two-asset portfolio (could be bonds, stocks etc). The expected return on that portfolio R(e) is

R(e)= x(a)r(a) + x(b)r(b)

where
x(a) =percentage invested in asset a
r(a) = expected return of asset a
x(b) = percentage invested in asset b
r(b) = expected return of asset b

The expected return volatility of that portfolio, σ(p) is given by 

 σ(p)²= x(a)²σ(a)² + x(b)²σ(b)² + 2x(a)x(b)σ(a)σ(b)ρ(ab)

where
σ(a) = Standard deviation of asset a
σ(b) = Standard deviation of asset b
ρ(ab) = Correlation between assets a and b

To be clear σ(n)² is the variance of asset n. The term σ(a)σ(b)ρ(ab) is termed the covariance of the portfolio. 

Example: take two equally weighted assets, with expected returns of 12% and 6%, return volatility of 12% and 6%
Set x(a) =50%, x(b) = 50%, r(a) = 12%, r(b) = 6%, σ(a) = 12%, σ(b) = 6%

The expected return R(e) = 9.00%
If the correlation is 100%, then the expected portfolio return volatility σ(p) = 9.00%. If the correlation is 0%, then σ(p) = 6.71%. If the correlation is -100%, then σ(p) = 3.00%.

So to reduce your portfolio return volatility, you want to find assets with as low a return correlation as possible. The correlation is far more important than the number of assets. You can add as many assets as you like but if the return correlation is 90% you will get little diversification benefit. Better to find a few uncorrelated assets. 

I spent some time last night talking to a six year old girl called Zoe, who went star gazing last month on the South Downs.  We went from there being 1 of her, to 100 in her school, to 7,000,000 in London, to 70,000,000 in the UK, to 7,000,000,000 people on the planet, to 100,000,000,000 stars "like the sun" in the Milky Way, to their being 100,000,000,000 galaxies in the universe "like the Milky Way" that we can see.

She was packed her off to bed with wonder in her eyes .

Are you sure her eyes had not just glazed over with all those unrelatable numbers ?